1. Technical Field
The disclosure relates to particle system analysis. More specifically, the disclosure relates to systems and methods that involve the use of polarization characteristics of light for analyzing particle systems.
2. Description of the Related Art
The polarization state of light is often described in relation to a plane of incidence defined by the vectors drawn in the direction of the incident and scattered rays. The transverse electric (TE) polarization refers to the polarization state of the light whose electric field vectors oscillate transverse, or perpendicular to this plane of incidence. The transverse magnetic (TM) polarization refers to the polarization state of the light whose magnetic field vectors oscillate transverse, or perpendicular to this plane of incidence. When a complex particle system having no preferentially oriented structure is illuminated by unpolarized light, the polarization state of the scattered light in the near-backward direction may be biased in favor of the TM state. Because fundamental physical processes like dipole scattering and Fresnel reflections favor the TE polarization state, this state is perhaps more commonly seen. A dipole may be described as a sphere whose total dimension is much smaller than the wavelength λ of the incident light. A bias toward the TM state is referred to as a negative polarization, and this negative polarization in the near-backward-scatter (˜<10 degrees from exact back-scatter) is sometimes referred to as the “Polarization Opposition Effect” (POE). On the basis of symmetry considerations, we expect the polarization in the exact back-scatter direction to be zero.
There appears to be two branches to the POE. One branch manifests itself as an asymmetric dip in the linear polarization state at approximately or less than 1 degree from the exact back-scatter direction. Researchers observing astronomical bodies have observed that the minimum position of this branch is approximately the width of the photometric opposition effect, i.e., the peak in the total intensity in the exact back-scatter direction produced by the coherent back-scattering mechanism, as described by V. K. Resonebush, V. V. Avramchuk, A. E. Rosenbush, and M. I. Mishchenko, “Polarization properties of the Galilean satellites of Jupiter: observations and preliminary analysis,” Astrophys. J. 487, 402–414 (1997), which is incorporated by reference herein. The photometric opposition effect refers to an increase in the absolute intensity of the scattered light in the backscattered direction and the maximum of this increase is located in the exact backward-scattering direction.
Another branch appears to be symmetric and parabolically shaped and is located at larger scattering angles, approximately but not limited to 5°–20° from the exact backward-scattering direction. It has been observed that one or both of these negative polarizations may be present in the same scattering system. As used herein, both branches of negative polarization phenomena are referred to as the polarization opposition effect (POE). Additional background information on the POE is provided, for example, by Muinonen, Shkuratov, et al., and the references cited therein. K. Muinonen, “Coherent back-scattering by solar system dust particles,” in Asteroids, Comets and Meteors, A. Milani, M. Di Martino, and A. Cellino, eds. (Kluwer, Dordrecth, the Netherlands 1974), pp. 271–296; Yu Shkruatov, A. Ovcharenko E. Zubko, V. Kaydash, D. Stankevich, V. Omelchenko, O. Miloslavskaya, K. Muinonen, J. Piironen, S. Kaasalaienen, R. Nelson, W. Smythe, V. Rosenbush and P. Helfenstein, “The opposition effect and negative polarization of structural analogs of planetary regoliths,” Icarus 159, 396–416 (2002), each of which is incorporated herein by reference.
The asymmetric branch of the POE has been inextricably linked with other phenomena, such as coherent back-scattering enhancement. See, M. I. Mishchekno, “On the nature of the polarization opposition effect exhibited by Saturn's rings,” Astrophys. J. 411, 351–361 (1993), which is incorporated herein by reference.
When a light ray traverses through a random medium to a detector, it is accompanied by another ray striking all the elements of the system in reverse order. When the detector is in the exact back-scatter direction, these rays constructively interfere because they have traversed the same path length but in reverse directions. When the detector is not in the exact back-scatter direction, the path lengths of these two reciprocal rays are no longer identical, and we see the intensity drop off because the constructive interference condition is optimized. The width of this peak is inversely proportional to the difference in path length between these two rays.
In a flourish of research along parallel lines, various researchers were able to show that the coherent back-scatter mechanism was also responsible for the asymmetric branch of the POE. Earlier approaches considered rigorous methods applicable to specific scattering systems, such as use of the vector theory of coherent back-scattering for a semi-infinite medium of the Rayleigh particles, V. D. Ozrin, “Exact solution for the coherent back-scattering of polarized light from a random medium of Rayleigh scatterers,” M. I. Mishchenko, “Polarization effects in weak localization of light: calculation of the copolarized and depolarized back-scattering enhancement factors,” Phys. Rev. B 44, 12579–12600 (1991); M. I. Mishchenko, “Enhanced back-scattering of polarized light from discrete random media,” J. Opt. Soc. Am. A 9, 978–982 (1992); M. I. Mishchenko, J. J. Luck and T. M. Nieuwenhuizen (hereinafter Mishchenko et. al.), “Full angular profile of the coherent polarization opposition effect,” J. Opt. Soc. Am. A 17, 888–891, (2000) or scattering from small particles near a surface, I. V. Lindell, A. H. Sihovla, K. O. Muinonen, and P. W. Barber, “Scattering by a small object close to an interface. I. Exact image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991) K. O. Muinonen, A. H. Sihvola, I. V. Lindell, and K. A. Lumme, “Scattering by a small object close to an interface. I. Study of back-scattering,” J. Opt. Soc. Am. A 8, 477–482 (1991), each of which is incorporated by reference herein.
More recent approaches applied Monte-Carlo-type ray-tracing computations for a generated particle system, involved keeping track of the phase of the scattered rays (K. Muinonen, “Coherent back-scattering by absorbing and scattering media,” in Light Scattering by Nonshperical Particles, B. Gustafson, L. Kolokolova, and G. Videen, eds. (U.S. Army Research Laboratory, Adelphi, Md., 2002) 223–226, which is incorporated herein reference herein) or involved an approximate expression for the scattering by a population of scatters. See, Yu. Shkuratov, M. Kreslavsky, A. Ovcharenko, D. Stankevich, E. Zubko, C. Pieters, and G. Arnold, “Opposition effect from Clementine data and mechanisms of back-scatter,” Icarus 141, 132–155 (1999), which is also incorporated by reference herein. Although these methods are significantly different in approach, there are significant similarities in their results. Part of the reason for this is that the polarization state of light scattered from a Rayleigh particle and from a surface facet are similar.
Because the two branches of the POE have significantly different shapes and one or both may be present in a scattering system, there is some debate as to the underlying physical mechanism of the symmetric, parabolically shaped branch. This was the subject of several discussions at a North Atlantic Treaty Organization Advance Research Workshop on the “Optics of Cosmic Dust” held in Bratislava, Slovakia, 16–19 Nov. 2001. Some researchers argued that the coherent back-scatter mechanism can explain the observations of both asymmetric and symmetric branches of the POE, whereas others did not believe that the evidence as yet presented provides an adequate proof of the mechanism. It appears that the source of some of the confusion is that, in the calculations of Mishchenko, et al., the POEs for a population of Rayleigh scatterers have the same asymmetric shape regardless of the mean free path. In fact, the polarization is even plotted as a function of a dimensionless parameter q=klγ, where l is the mean free path (or average distance a ray will travel before interacting with another surface dipole), k is the spatial frequency defined in terms of wavelength λ as k=2π/λ, and γ is the scattering angle; hence the angular minimum can be found directly from this plot as       q    min    ≈            1.68      kl        .  
Other methodologies that are able to produce a more symmetric, parabolic branch at larger scattering angles are either approximate techniques or ones in which the physical mechanisms are not as transparent. The presence of sometimes one and sometimes both branches measured from the same object has served to add to the confusion. Adding fuel to the fire is the experimental research of Geake and Geake; J. E. Geake and M. Geake, “A remote-sensing method for sub-wavelength grains on planetary surfaces by optical polarimetry,” Mon. Notes R. Astron. Soc. 245, 46–55 (1990), which is incorporated herein by reference. They discovered that the angular minimal positions measured from the back-scatter of their samples increase with particle size parameter. This is exactly the opposite of what would be predicted by a coherent back-scatter mechanism. Much of the current understanding of the POE is contained in four chapters (astronomical observations, laboratory measurements, theory, and numerical techniques) of a book written by participants of the aforementioned workshop; specifically, G. Videen and M. Kocifaj, eds., “Optics of Cosmic Dust” (Kluwer Academic, Dordrecht, The Netherlands, 2002), which is incorporated by reference herein.